Question

Do rozwiązania zaznaczone zadania w załączniku :)

Answer 1

3.
a)
 √(13 - 4√3) + √(4(7 - 4√3)) = 3

√(13 - 4√3) + √(4(7 - 4√3)) = √(13 - 4√3) + 2√(7 - 4√3) =

= √(13 - 4√3) + 2√(7 - 4√3) = √(12 - 2√12 + 1) + 2√(3 - 4√3 + 2^2) =

= √(√12 -1)² + 2√(2 - √3)² = √12 -1 + 2(2 - √3) = 2√3 -1 + 4 - 2√3 =

= 4 - 1 = 3
b)
√(9 - 4√2) + 1 = 4/√2

√(9 - 4√2) + 1 =√(18/2 -(8/2)√2) + 1 = √[(18 -8√2)/2] + 1 =
 
= [√(16 - 8√2 + 2)]/√2 + 1 =

= [√(4² - 2*4*√2 + (√2)²)]/√2 + 1= [√(4 - √2)^2]/√2 + 1 =

= (4 - √2)/√2 + 1 = 4/√2 - 1 + 1 = 4/√2

c)
√(6 -2√5) + 2(5^(1/4) + 1)) = (1 + 5^(1/4))²

√(6 -2√5) = √(5 -2√5 + 1) = √((√5)² -2*1*√5 + 1²) = √(√5 - 1)² = √5 - 1

√5 - 1 + 2*(5^(1/4)) +2  = √5 + 2*(5^(1/4)) + 1 = [5^(1/4)]²+ 2*(5^(1/4))+1² =

= (1 + 5^(1/4))²

d)
(√2 - 1)/(√2 + 1) = [(5√2 - 7)/(5√2 + 7)]^1/3

(√2 - 1)³ = (√2)³ - 3*(√2)² + 3*(√2) - 1 = 2√2 - 6 + 3√2 - 1 = 5√2 - 7

(√2 + 1)³ = (√2)³ + 3*(√2)² + 3*(√2) + 1 = 2√2 + 6 + 3√2 - 1 = 5√2 + 7

4)
a)
6/(3 - 2√3) = 6(3 + 2√3)/(3 + 2√3)(3 - 2√3) = 6(3 + 2√3)/(9 - 12) = - 2(3 + 2√3)
b)
√6/(2√3 + √2) = (2√3 - √2)√6/((2√3)² - (√2)²)=(2√3 - √2)√6/(12-2) = 0,1*(2√3 - √2)√6
c)
3/(1 + ³√2) = 3(1-³√2 +(³√2)²)/(1³+(³√2)³) =3(1-³√2 +³√4)/3 = 1 - ³√2 + ³√4
d)
1/(³√3 - ³√2)=((³√3)²+(³√3)*(³√2)+(³√2)²)/((³√3)³ - (³√2)³) =

= (³√9 + ³√6 + ³√4)/(3 - 2) = ³√9 + ³√6 + ³√4

e)
2/(√2 + √3 -1) = 2(√2 + √3 +1)/(√2 + √3 -1)(√2 + √3 +1)

(√2 + √3 -1)(√2 + √3 +1) = (√2 + √3)² -1 = 2+3 -1 + 2√6 = 4+2√6

2/(√2 + √3 -1) = 2(√2 + √3 + 1)(4 - 2√6)/(4 + 2√6)(4 - 2√6)

(4 + 2√6)(4 - 2√6) = 16 - 24 = -8

2/(√2 + √3 -1) = 2(√2 + √3 +1)(4 - 2√6)/(-8)

(4 - 2√6)(√2 + √3 +1) = 4√2 + 4√3 + 4 - 2√12 - 2√18 - 2√6 =

=4√2 + 4√3 + 4 -4√3 - 6√2 - 2√6 = -2√2 + 4 - 2√6

2/(√2 + √3 -1) = 2(-2√2 + 4 -2√6)/(-8) = 0,5(√2 + √6 - 2)